M. S. Mahadeva Naika, S. Chandankumar and N. P. Suman (Received August 6, 2012)

Transcription

1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume , CERTAIN NEW MODULAR IDENTITIES FOR RAMANUJAN S CUBIC CONTINUED FRACTION M. S. Mahadeva Naika, S. Chandankumar and N.. Suman Received August 6, 2012 Abstract. In this paper, first we establish some new relations for ratios of Ramanujan s theta functions. We establish some new general formulas for explicit evaluations of Ramanujan s theta functions. We also establish new relations connecting Ramanujan s cubic continued fraction V q with four other continued fractions V q 15, V q 5/, V q 21 and V q 7/. 1. Introduction On page 66 of his lost notebook [14], S. Ramanujan gave the cubic continued fraction V q := q1/ 1 q q 2 1 q 2 q 4 1 q q 6..., q < and claimed that there are many results of V q analogous to those for the Rogers- Ramanujan continued fraction. H. H. Chan [5] has established all the claims made by Ramanujan. Subsequently, many mathematicians have contributed to the theory of Ramanujan s cubic continued fraction. Some of them are N. D. Baruah [2], C. Adiga, T. Kim, M. S. Mahadeva Naika and H. S. Madhusudhan [1], Mahadeva Naika [7], B. Cho, J. K. Koo and Y. K. ark [6]. Recently, Mahadeva Naika, S. Chandankumar and K. Sushan Bairy [10], [11] have established new modular identities connecting V q with V q n, for n = 4, 6, 8, 9, 10, 12, 14, 16, 17, 19 and 22. In [15], J. Yi introduced two parameters h k,n and h k,n as follows: h k,n := ϕeπ n/k k 1/4 ϕe π nk, 1.2 h k,n := ϕeπ n/k k 1/4 ϕe π nk 1. and established several properties as well as explicit evaluations of h k,n and h k,n for different positive rational values of n and k. Recently, Mahadeva Naika and Chandankumar [9] have established several new modular equations of degree 2 and 2010 Mathematics Subject Classification D10, 11A55, 11F27. Key words and phrases: Cubic continued fraction, Modular equations, Theta function. The authors are grateful to Dr. Shaun Cooper and the referee for helpful comments and suggestions.

2 158 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND N.. SUMAN established general formulas for explicit evaluations of h 2,n. They have also established several new explicit evaluations for the Ramanujan-Göllnitz-Gordon continued fraction, the Ramanujan-Selberg continued fraction and a continued fraction of Eisenstein. In [8], Mahadeva Naika, Bairy and M. Manjunatha have established several new modular equations of degree 4 and established general formulas for explicit evaluations of h 4,n. In [], Baruah and Nipen Saikia and Yi, Y. Lee and D. H. aek [16] have defined two parameters l k,n and l k,n as follows: and ψe πn/k l k,n := 1.4 k 1/4 e k1π 8 n/k ψe π nk, l k,n ψe πn/k := 1.5 k 1/4 e k1π 8 n/k ψe π nk. They have also established several properties as well as explicit evaluations of l k,n and l k,n for different positive rational values of n and k. In [12], Mahadeva Naika, Chandankumar and Bairy have established several new modular equations of degree 9 and also established several general formulas for explicit evaluations for the ratios of Ramanujan s theta function ψ. Consider ϕq := fq, q = q n2, 1.6 ψq := fq, q = n= q nn1/2, 1.7 n=0 which are special cases of Ramanujan s general theta function fa, b := a nn1/2 b nn1/2, ab < 1. Let Kk := π 2 0 n= dφ 1 k2 sin 2 φ = π 2 n= n n! 2 k2n = π F 1 2, 1 ; 1; k2, where 0 < k < 1 and 2 F 1 is the ordinary or Gaussian hypergeometric function defined by a 2F 1 a, b; c; z := n b n z n, 0 z < 1, c n n! n=0 where a 0 = 1, a n = aa 1 a n 1 for n a positive integer with a, b and c are complex numbers such that c 0, 1, 2,.... The number k is called the modulus of K and k := 1 k 2 is called the complementary modulus. Let K, K, L and L denote the complete elliptic integrals of the first kind associated with the moduli k, k, l and l, respectively. Suppose that the equality n K K = L L 1.9

3 MODULAR IDENTITIES FOR RAMANUJAN S CUBIC CONTINUED FRACTION 159 holds for some positive integer n. Then a modular equation of degree n is a relation between the moduli k and l which is induced by 1.9. Following Ramanujan, set α = k 2 and β = l 2. Then we say β is of degree n over α. The multiplier m is defined by m = K L In Section 2, we collect some identities which are useful to prove our main results. In Section, we establish some new modular equations of degree for the ratios of Ramanujan s theta functions. In Section 4, we establish some general formulas for the explicit evaluations of h,n and l,n, for positive rational values of n. In section 5, we establish modular relations for V q with four other continued fractions V q 15, V q 5/, V q 21 and V q 7/. 2. reliminary Results In this section, we record identities which are useful in proving our main results. Lemma 2.1. [4, Ch. 17, Entry 10 i, Entry 11 ii, pp ] For 0 < α < 1, ϕq = z, 2.1 2q 1/8 ψq = z{α1 α} 1/8, 2.2 where z := 2 F 1 1 2, 1 2 ; 1; α. Lemma 2.2. [4, Ch. 20, Entry 11 viii, ix, p. 84] Let α, β, γ and δ be of the first, third, fifth and fifteenth degrees respectively. Let m denote the multiplier connecting α and β and m be the multiplier relating γ and δ. Then 1/8 1/8 1/8 αδ 1 α1 δ αδ1 α1 δ m = βγ 1 β1 γ βγ1 β1 γ m, 2. 1/8 βγ αδ 1/8 1 β1 γ 1 α1 δ 1/8 βγ1 β1 γ m = αδ1 α1 δ m. 2.4 Lemma 2.. [4, Ch. 20, Entry 1 i, ii, p. 401] Let α, β, γ and δ be of the first, third, seventh and twenty first degrees respectively. Let m denote the multiplier connecting α and β and m be the multiplier relating γ and δ, then 1/4 1/4 1/4 βγ 1 β1 γ βγ1 β1 γ αδ 1 α1 δ αδ1 α1 δ 1/6 2.5 βγ1 β1 γ 4 = m αδ1 α1 δ m, 1/4 1/4 αδ 1 α1 δ αδ1 α1 δ βγ 1 β1 γ βγ1 β1 γ 1/6 αδ1 α1 δ 4 = m βγ1 β1 γ m. 1/4 2.6

4 160 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND N.. SUMAN Lemma 2.4. [4, Ch. 20, Entry 1 i, p. 45] We have where V q is defined as in the equation 1.1. Lemma 2.5. [4, Ch. 20, Entry 1iii, p. 45] We have Lemma 2.6. [1] If M := ϕq 1/ ϕq ϕq 9 ϕq 1 1 V q = ψ4 q qψ 4 q, 2.7 ϕ 4 1/ q = 1 ϕ 4 q 1, 2.8 9ϕ 4 q 1/ = 1 ϕ q ψq q 1/4 ψq and N := ϕq ϕq, then N 4 M 4 N 4 = 9 M Modular Equations of Degree Three In this section, we establish several new modular equations for the ratios of Ramanujan s theta functions ϕ and ψ. Theorem.1. If := ϕq ϕq and Q := ϕq15 ϕq 45, then Q 9 9 = 1860Q Q 0 Q 20 Q7 7 7 Q Q 5 2 Q [ 15 Q Q 241Q 15 5 Q 5 296Q5 5 [ 180 Q 107Q Q Q Q 154Q [ 11 Q 140Q 4 Q Q 98 Q 179Q ] 5 2 Q Q Q 5 1Q5 5 Q 22 Q 2 5 ] [ Q 5 20Q Q Q 2Q 15 ] [ Q 28Q 5 5 Q Q 5 Q 18Q ] Q Q 6 Q Q 6 9 Q 6 Q Q Q 5 50 Q6 8 Q8 6. ].1

5 MODULAR IDENTITIES FOR RAMANUJAN S CUBIC CONTINUED FRACTION 161 roof. Transcribing the equations 2. and 2.4 by using the equations 2.1 and 2.2 into theta function, we deduce that a q 1 = b q 1,.2 where := ϕq ϕq, q 1 := ϕq5 ϕq 15, a := ψq q 1/4 ψq, b := ψq5 q 5/4 ψq 15. Using the equation 2.10 in the above equation.2 along with the equation 2.9, we deduce that 9Q 6 6Q 6 4 A 2 15Q 6 4 A 15Q 5 A 2 45Q 4 2 A 27Q A Q 5 7 A 2 90Q 2 4 A Q 6 8 A Q 6 8 A 2 27Q 7 A 2 0Q 4 6 A 2 48Q 5 A 15Q 4 6 A 45Q 2 4 A 2 18Q Q Q 12Q 5 15Q Q Q 7 2Q Q 5 7 = 0,. where A := Solving the above equation. for A and then cubing both sides, we arrive at.1. Remark 1. The equation.1 holds for := ψq q 1/4 ψq and Q := ψq15 q 15/4 ψq 45. Eliminating and q 1 in the equation.2, we arrive at the equation involving a and b. Replacing q by q in the resulting equation, the Remark 1 holds.

6 162 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND N.. SUMAN Theorem.2. If := ϕq ϕq and Q := ϕq5/ ϕq 5, then 9 20 Q Q Q 7 27Q Q Q 21Q Q [ Q Q 98Q Q 5 15Q Q Q [ 140 Q 11Q 4 Q Q 4 5 Q Q 5 8 Q Q 8 ] 2 Q [ 10 2 Q 12Q 2 Q 2 [ Q 180Q Q Q 126Q ] Q Q Q5 Q5 5 4 Q Q [ 5 18 Q Q 15 Q Q 15 Q5 2 Q5 5 ] ] Q 15Q ] Q Q 50 7 Q Q 7 = Q 6 Q 8.4 roof. The proof of the equation.4 is similar to the proof of the equation.1 except that in the place of the equation 2.9; equation 2.8 is used. Remark 2. The equation.4 holds for := ψq q 1/4 ψq and Q := ψq5/ q 5/12 ψq 5. Theorem.. If := ϕq ϕq and Q := ϕq21 ϕq 6, then Q 12 = 6Q Q Q 7 200Q7 7 Q 4 754Q Q Q2 2 9 Q Q 9 21 Q Q Q Q Q Q Q 6 41Q Q 4621Q Q Q 54 5 Q Q Q 5 2 Q 2 [ Q Q Q Q 12 Q Q 1628Q 9 Q

7 MODULAR IDENTITIES FOR RAMANUJAN S CUBIC CONTINUED FRACTION [ [ Q 2 98Q2 2 Q 5 551Q5 5 Q 8 269Q Q 106Q Q 109Q Q 6 15Q6 6 7 Q 22Q Q 5 7Q Q 949Q Q 4 56Q Q 6 107Q ] 4 Q Q Q 2 29Q Q 7 215Q7 7 Q 4 21Q4 4 Q 5 28Q ] Q 7 85Q Q Q 6 2 Q 2 Q Q 4 45Q4 4 8 Q 8 8 ] Q 8 [ 7 Q 6Q ] 6.5 roof. The proof of the equation.5 is similar to the proof of the equation.1 except that in the place of the equations 2. and 2.4; equations 2.5 and 2.6 are used. Remark. The equation.5 holds for := ψq q 1/4 ψq and Q := ψq21 q 21/4 ψq 6. Theorem.4. If := ϕq ϕq and Q := ϕq7/ ϕq 7, then = 6 Q Q Q10 6 Q Q Q8 5 Q8 8 Q7 567 Q Q 6 1Q6 6 Q 5 279Q Q Q Q 96Q Q 2 15 Q Q 18Q 21 Q Q Q Q Q 4 Q Q Q Q Q Q 6 Q 9 11 Q Q 11 2 Q Q 2

8 164 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND N.. SUMAN [ Q 99Q Q 4 19Q Q 7 78Q7 7 [ 4 Q Q 4 14 Q 2 5Q2 2 Q 5 459Q Q 8709Q Q8 5 Q Q 267Q Q 4 29Q ] Q 6 7Q Q Q Q 2 Q Q 4 7Q [ 8 Q Q Q Q ] Q 12Q Q6 2 Q6 6 ] Q 2 11Q [ 147 Q 5 Q5 5 Q 5 6Q Q 7Q ] roof. The proof of the equation.6 is similar to the proof of the equation.5 except that in the place of the equation 2.9; equation 2.8 is used. Remark 4. The equation.6 holds for := ψq q 1/4 ψq and Q := ψq7/ q 7/12 ψq Formulas for Explicit Evaluations of Ramanujan s Theta Functions In this section, we establish several new general formulas for explicit evaluations of ratios of Ramanujan s theta functions by using the modular equations of degree established in the previous section. Theorem 4.1. If := h,n and Y := h,225n, then Y 9 9 = 1860 Y 5 20 Y Y Y 21 Y Y 0 Y 1 [ Y Y Y 179 Y [ 15 Y 241 Y 180 Y 107 Y Y Y 5 Y 154 Y Y 1 [ Y 11 Y 140 Y ] 2 Y Y 2 Y 5 1 Y 5 ] Y 22 Y

9 MODULAR IDENTITIES FOR RAMANUJAN S CUBIC CONTINUED FRACTION Y 5 20 Y 5 Y 28 Y ] 2 4 Y Y 4 ] 9 Y 5 1 ] 27 Y Y Y Y 5 6 Y Y 5 8 Y 6 Y 6 8 Y 8 6 [ Y 2 Y [ roof. Using the equations.1 and 1.2, we arrive at 4.1. Y 18 Y 9 Y 1 Y Lemma 4.1. We have h,15 = /4 2, /4 5 h,1/15 =, 4. l,15 = /4 2, /4 l,1/15 =. 4.5 roofs of 4.2 and 4.. utting n = 1/15 in the equation 4.1 and using the fact that h,15 h,1/15 = 1, we deduce that h 8,15 186h 4, h 4, h 8,15 6h 4,15 2 h 4,15 2 h 4, = Since the first factor of the equation 4.6 is zero for the specific value of q = e π/5, whereas the other factors are not zero, we find that h 8,15 186h 4, h 4, = On solving 4.7 and using 0 < h,15 < 1, we arrive at 4.2. Since h,15 h,1/15 = 1, we arrive at 4.. roofs of 4.4 and 4.5. Using the equations 2.10, 4.2 and 4., we arrive at 4.4 and 4.5.

10 166 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND N.. SUMAN Theorem 4.2. If := h,n and Y := h,25n/9, then 9 20 Y Y 7 27 Y Y Y Y 1 [ Y Y 98 Y ] Y Y 5 15 Y 5 Y Y [ 140 Y 11 Y 2 4 Y Y 4 9 Y Y 6 6 Y 8 7 Y 5 1 Y Y 21 Y 2 Y 2 Y 5 Y 5 4 Y Y [ 10 2 Y 12 Y Y 5 27 = Y 241 Y 126 Y [ Y 180 Y ] 6 [ 5 18 Y Y 15 8 Y Y 8 Y 1 Y 20 5 Y 5 Y 5 ] 2 28 Y 15 Y ] Y Y ] Y Y roof. Using the equations.4 and 1.2, we arrive at 4.8. Lemma 4.2. We have h,5/ = /4 5, 4.9 h,/5 = /4, /4 5 l,5/ =, 4.11 l,/5 = / roofs of 4.9 and utting n = /5 in the equation 4.8 and using the fact that h,5/ h,/5 = 1, we deduce that h 8,5/ 6h4,5/ 2 h 4,5/ 12 h 4,5/ 2 2 9h 8,5/ 6h4,5/ 12 h 4,5/ 7 4 = Since the last factor of the above equation 4.1 is zero for the specific value of q = e π/5 and other factors are not zero, we find that 9h 8,5/ 6h4,5/ 12 h 4,5/ 7 4 = Solving the above equation 4.14 and h,5/ < 1, we arrive at 4.9 and 4.10.

11 MODULAR IDENTITIES FOR RAMANUJAN S CUBIC CONTINUED FRACTION 167 roofs of 4.11 and Using the equation 2.10 along with the equations 4.9 and 4.10 respectively, we arrive at 4.11 and Theorem 4.. If := h,n and Y := h,441n, then Y Y = Y 41 Y Y Y Y Y Y Y 6 41 Y Y Y Y Y Y 4621 Y Y Y Y Y Y Y 12 9 Y Y Y Y Y Y Y Y Y 2 1 [ 2 Y Y 1628 Y Y 2 98 Y Y 949 Y Y 4 56 Y Y Y Y Y Y Y Y Y 8 ] Y Y 4 [ 8709 Y 106 Y Y 2 29 Y Y 109 Y Y 4 21 Y Y 5 28 Y Y 6 15 Y Y 7 85 Y 7 ] Y Y 6 [ Y 22 Y Y 2 Y Y 4 45 Y Y 5 7 Y 5 ] Y 8 1 [ 8 Y 8 7 Y 6 Y ] roof. Using 1.2 and.5, we arrive at 4.15.

12 168 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND N.. SUMAN Lemma 4.. We have h,21 =, h,1/21 =, l,21 =, l,1/21 = roof of 4.16 and utting n = 1/21 in the equation 4.15 and using the fact that h,21 h,1/21 = 1, we deduce that h 12,21 0h 10,21 81h 8,21 60h 6,21 9h 4,21 18h 2, h 6,21 h 4,21 h 2,21 2 h 8,21 h 6,21 6h 2,21 = Since the last factor of the equation 4.20 is zero for the specific value of q = e π/7, we find that h 8,21 h 6,21 6h 2,21 = On solving 4.21 and using 0 < h,21 < 1, we arrive at 4.16 and roofs of 4.18 and Using the equation 2.10 along with the equations 4.16 and 4.17 respectively, we arrive at 4.18 and Theorem 4.4. If := h,n and Y := h,49n/9, then 12 = Y 12 Y Y 6 Y Y Y 8 Y Y 7 Y Y 6 1 Y Y Y Y Y Y 96 Y Y 2 15 Y Y 18 Y 1701 Y Y Y Y Y Y Y Y Y Y Y 1 Y Y Y 11 2 Y Y 2 [ Y 99 Y Y 2 5 Y Y 12 Y 4.22

13 MODULAR IDENTITIES FOR RAMANUJAN S CUBIC CONTINUED FRACTION Y 4 19 Y Y 7 78 Y Y Y Y 8 Y Y 6 2 Y 6 6 ] Y 4 1 [ 4 Y Y 8709 Y Y 2 11 Y Y 267 Y Y 4 29 Y Y 5 6 Y Y 6 7 Y 6 ] Y 6 1 [ 6 Y Y 7 Y Y 2 Y Y 4 7 Y Y 5 Y 5 ] Y 8 1 [ 8 Y Y Y ] 6. roof. Using.6 and 1.2, we arrive at Lemma 4.4. We have h,7/ =, h,/7 =, l,7/ =, l,/7 = roofs of 4.2 and utting n = /7 in the equation 4.22 and using the fact that h,7/ h,/7 = 1, we deduce that h 8,7/ 6h6,7/ h2,7/ 1h6,7/ h4,7/ h2,7/ 12 9h 12,7/ 18h10,7/ 9h8,7/ 60h6,7/ 81h4,7/ 0h2,7/ 12 = Since the first factor of the equation 4.27 is zero for the specific value of q = e π/ 7 and other factors are not zero, we find that h 8,7/ 6h6,7/ h2,7/ 1 = Solving 4.28 and using 0 < h,7/ < 1, we arrive at 4.2 and roofs of 4.25 and Using the equation 2.10 along with the equations 4.2 and 4.24 respectively, we arrive at 4.25 and 4.26.

14 170 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND N.. SUMAN 5. Modular Relations for Ramanujan s Cubic Continued Fraction In this section, we establish modular relation connecting Ramanujan s cubic continued fraction V q with each of V q 15, V q 5/, V q 7/ and V q 21. Theorem 5.1. If v = V q and w = V q 15, then w v 8192v v 12 20v v 9 w v v 49152v v v 9 21 w v 49920v v v v 15 w v v 9 204v v 15 21v w v v v v v 6 w v v v 4096v v v w v v v v v 12 90v w v v v v v v 15 w v v v v v 684v 6 w v v 25v v v v 15 w v v v v v 7800v 9 w v v v 4512v v v 6 w v 6 688v v v v 18 60v w v 60v 9 18v v v v 18 w 4 195v 6 200v 9 210v v v 15 21v w 650v 15 84v 18 10v 6 15v v 9 6v w 2 v 15v 6 90v 15 60v 9 12v 18 20v 12 w v 18 = roof. Using the Remark 1 along with the equations.1 and 2.7, we deduce that 27w 9 v w 12 v 9 225v 9 w 45v 6 w 447v 12 w 1728w 12 v 4650v 9 w 6 108w 9 v 6 60v 6 w v 12 w v 9 w w 12 v 6 w 9 6w v 12 w w 12 v w 12 v x 2 w 24 v 25280x 2 w 24 v x 2 w 15 v x 2 w 15 v 9 20x 2 w 15 v 6 45x 2 w 15 v 186x 2 w v x 2 w 12 v x 2 w 9 v x 2 w 6 v x 2 w 9 v 9 514x 2 w 12 v 6 27x 2 w 9 v 6 45x 2 w 12 v 196x 2 w 6 v 9 2x 2 w 6 v w 18 v 12 x b 2 w 12 v 15 x b 2 w 6 v 15 x b 2 w 18 v 15 x b 2 w 15 v 15 x 2 87w 18 v x w 9 v 15 x w 18 v 9 x w 12 v 9 x w 18 v 6 x 2 2w v 9 x 2 w v 6 x 2 590w v 15 x x 2 v 18 w x 2 v 18 w x 2 v 18 w 6 415x 2 v 18 w x 2 w 21 v 9

15 MODULAR IDENTITIES FOR RAMANUJAN S CUBIC CONTINUED FRACTION x 2 w 24 v x 2 w 24 v x 2 w 24 v x 2 w 24 v x 2 w 24 v x 2 w 12 v x 2 w 21 v x 2 w 18 v x 2 w 15 v x 2 w 18 v x 2 w 21 v x 2 w 21 v x 2 w 15 v w 21 x 2 v 6 441w 21 x 2 v x 2 v 21 w x 2 v 21 w 6 675x 2 v 21 w x 2 v 24 w x 2 v 24 w x 2 v 24 w 6 210x 2 v 24 w x 2 v 24 w x 2 v 24 w x 2 v 24 w x 2 v 24 w x 2 v 18 w 21 15w 15 6w 24 15w 21 20w w 15 v 48615w 15 v w 15 v w 15 v v 15 w v 18 w v 18 w v 18 w v 18 w 20145v 15 w v 15 w v 15 w w 24 v w 24 v w 21 v w 18 v w 12 v w 24 v w 21 v w 18 v w 15 v w 24 v w 21 v w 21 v w 18 v w 18 v w 15 v w 15 v w 24 v w 21 v w 18 v v 21 w v 21 w v 21 w w 24 v w 21 v w 18 v 6 699w 24 v 25785w 21 v 1590w 18 v 28464v 24 w v 24 w v 24 w 6 285v 24 w v 24 w v 24 w v 24 w v 24 w w 24 v 27 18x 2 w 27 v 0x 2 v 27 w 960x 2 w 27 v x 2 w 27 v x 2 w 27 v x 2 w 27 v x 2 w 27 v x 2 w 27 v x 2 w 27 v x 2 v 27 w x 2 v 27 w x 2 v 27 w 9 16x 2 v 27 w x 2 v 27 w a 2 v 27 w 24 b a 2 v 27 w 21 b a 2 v 27 w 18 b w 27 v w 27 v w 27 v w 27 v w 27 v w 27 v w 27 v w 15 v v 27 w w 18 v 27 w 27 v w 27 v w 27 v 402v 27 w v 27 w w 21 v 27 = 0, 5.2 where x := ψq ψq 15 q 4 ψq ψq 45. Collecting the terms containing x 2 on one side of the equation 5.2 and then squaring both sides, we arrive at 5.1. This completes the proof.

16 172 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND N.. SUMAN Theorem 5.2. If v = V q and w = V q 5/, then v w 12 20w w w 9 180w v w w w w w 15 v w 840w w w w 9 v w w 9 204w w 15 21w v w w w w 10420w 12 6 v w w w w w 6 564w v w w w w 18 90w 1200w 9 v w w w w 6 65w 10140w 9 v w w w w 6 45w w 15 v w w w w 10140w w 6 v w w w w w 15 28w 6 v w w w w w w v w 18 60w 6975w w 6 688w w 9 v w w w 15 11w 18w 6 60w 9 v 4 200w 9 21w 195w 6 210w w w 18 v 15w 12 10w 6 6w 84w w w 9 v 2 12w 18 90w 15 15w 6 w 60w 9 20w 12 v w 18 = roof. The proof of the equation 5. is similar to the proof of the equation 5.1; except that in the place of the equation.1, the equation.4 is used. Theorem 5.. If v = V q and w = V q 21, then v 24 w v v v v v v v w v v v v v v v 15 w v v v v v v v w v v v v v v v 18 w v 87712v v v 9 068v v v 6 w v v v v v 81944v v 15 w v v v v v v v 21 w 17

17 MODULAR IDENTITIES FOR RAMANUJAN S CUBIC CONTINUED FRACTION v v v v v v v v 21 w v v v v v v v v 21 w v v v v v v 2076v v 9 w v 12 1v 1469v v v v v v 9 w v 21 24v 16799v v v v v v 12 8 w v v v v v v v v w v v v v v v v v 21 w v v v v v v v v 24 w v v v v v v v v 18 w v 4587v v v v v v v 18 w v v 9 111v v 17176v v 15 50v v 24 w v v v 18 2v 949v 12 6v v v 24 w v v v v 9 17v 465v 6 191v v 15 w v v 12 66v v 21 6v 896v v 6 616v 18 w v 18 72v v v 9 21v 6 251v 12 4v 1260v 21 w 2 147v 9 84v v 21 16v v 18 21v 6 4v 12 v w = roof. The proof of the equation 5.4 is similar to the proof of the equation 5.1; except that in the place of the equation.1, the equation.5 is used. Theorem 5.4. If v = V q and w = V q 7/, then w 24 v w w w w w 9 84w 252w 6 1 v w w w w w 2 20 w w 12 9 v w w w w w 616w w 18 v w w w w w w w 21 v 20

18 174 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND N.. SUMAN w 6 068w w w w w w v w w w w w w w v w w w w w w w 18 v w w w w w w w w 6 v w w w w w w w w 6 v w w w w w w w w 6 v w 6 552w w w w 15 1w w w 21 v w 28024w w w w w w w 18 v w w w w w 9255w w w 9 v w w w w w w w 5761w 6 9 v w w w w w w w w 24 v w w w w 2892w w w w 6 v w w w w w w 15 92w 4587w 9 v w w w w 24 50w w 9 108w 40972w 15 v w 18 6w w w 9 949w w 15 2w 14848w 21 v w w w 9 608w w w 15 17w 26520w 18 v 4 896w w w 6 6w 66w w w w 18 v w w 9 251w 12 21w w w 18 72w 24 4w v 2 4w 12 w 147w 9 588w w 21 21w 6 84w 15 16w 24 v = roof. The proof of the equation 5.5 is similar to the proof of the equation 5.; except that in the place of the equation.4, the equation.6 is used. References [1] C. Adiga, T. Kim, M. S. Mahadeva Naika and H. S. Madhusudhan, On Ramanujan s cubic continued fraction and explicit evaluations of theta-functions, Indian J. ure Appl. Math., , [2] N. D. Baruah, Modular equations for Ramanujan s cubic continued fraction, J. Math. Anal. Appl., ,

19 MODULAR IDENTITIES FOR RAMANUJAN S CUBIC CONTINUED FRACTION 175 [] N. D. Baruah and N. Saikia, Two parameters for Ramanujan s theta functions and their explicit values, Rocky Mountain J. Math., , [4] B. C. Berndt, Ramanujan s Notebooks, art III, Springer-Verlag, New York, [5] H. H. Chan, On Ramanujan s cubic continued fraction, Acta Arith., , [6] B. Cho, J. K. Koo and Y. K. ark, On the Ramanujan s cubic continued fraction as modular function, Tohoku Math. J., , [7] M. S. Mahadeva Naika, Some theorems on Ramanujan s cubic continued fraction and related identities, Tamsui Oxf. J. Math. Sci., , [8] M. S. Mahadeva Naika, K. Sushan Bairy and M. Manjunatha, Some new modular equations of degree four and their explicit evaluations, Eur. J. ure Appl. Math., , [9] M. S. Mahadeva Naika, S. Chandankumar, Some new modular equations and their applications, Tamsui Oxf. J. Inf. Math. Sci., , [10] M. S. Mahadeva Naika, S. Chandankumar and K. Sushan Bairy, New identities for Ramanujan s cubic continued fraction, Funct. Approx. Comment. Math., 46 1, 2012, [11] M. S. Mahadeva Naika, S. Chandankumar and K. Sushan Bairy, On some new identities for Ramanujan s cubic continued fraction, Int. J. Contemp. Math. Sci., , [12] M. S. Mahadeva Naika, S. Chandankumar and K. Sushan Bairy, Modular equations for the ratios of Ramanujan s theta function ψ and evaluations, New Zealand J. Math., , 48. [1] S. Ramanujan, Notebooks 2 Volumes, Tata Institute of Fundamental Research, Bombay, [14] S. Ramanujan, The Lost Notebook and Other Unpublished apers, Narosa, New Delhi, [15] J. Yi, Theta-function identities and the explicit formulas for theta-function and their applications, J. Math. Anal. Appl., , [16] J. Yi, Y. Lee and D. H. aek, The explicit formulas and evaluations of Ramanujan s theta-function ψ, J. Math. Anal. Appl., , M. S. Mahadeva Naika, S. Chandankumar and N.. Suman Department of Mathematics, Bangalore University, Central College Campus, Bangalore , Karnataka, India msmnaika@rediffmail.com, chandan.s17@gmail.com, npsuman.1@gmail.com